Comparison of the X-ray luminosity function (XLF) predicted by the bivariate distribution Ψ(M_⋆,λ_SAR,z) with direct observations. The blue solid line shows the XLF derived from our default model (A), the red dashed line is for the best-fit M_⋆-independent SARDF model (B), and the green solid line is for the best fit without the XLF into the likelihood function (no XLF). The black circles show the binned XLF for our XMM-COSMOS sample using the V_max method, indicating the luminosity range covered by XMM-COSMOS. The shaded cyan and yellow areas show the XLF by Miyaji et al. (2015) and Ueda et al. (2014), respectively, where the shaded area includes the variation of the XLF over the redshift range of the bin and the uncertainty of the XLF determination.